Let `f:R to R` be a function defined by `f(x)=(x^(2)-8)/(x^(2)+2)`. Then f is

Options

A. one-one but not onto
B. one-one and onto
C. one but not one-one
D. neither one-one nor onto

Answer

To determine the nature of the function \( f(x) = \frac{x^2 - 8}{x^2 + 2} \), we need to analyze whether it is one-to-one (injective) and onto (surjective).

### Step 1: Check if the function is one-to-one (injective)

A function is one-to-one if different inputs produce different outputs. To check this, we can evaluate \( f(x) \) for \( x \) and \( -x \):

1. Calculate \( f(1) \):
\[ f(1) = \frac{1^2 - 8}{1^2 + 2} = \frac{1 - 8}{1 + 2} = \frac{-7}{3} \]

2. Calculate \( f(-1) \):
\[ f(-1) = \frac{(-1)^2 - 8}{(-1)^2 + 2} = \frac{1 - 8}{1 + 2} = \frac{-7}{3} \]

Since \( f(1) = f(-1) \) but \( 1 \neq -1 \), the function is not one-to-one.

### Step 2: Check if the function is onto (surjective)

A function is onto if every possible output in the codomain is achieved by some input in the domain. The codomain given is \( \mathbb{R} \).

To check if the function can take the value \( 1 \):
1. Set \( f(x) = 1 \):
\[ \frac{x^2 - 8}{x^2 + 2} = 1 \]

2. Cross-multiply:
\[ x^2 - 8 = x^2 + 2 \]

3. Simplifying gives:
\[ -8 = 2 \]
which is a contradiction.

Since \( f(x) \) can never equal \( 1 \), the function does not cover all real numbers, meaning it is not onto.

### Conclusion

Since the function \( f(x) \) is neither one-to-one nor onto, we conclude that:

- The function is many-one (not one-to-one).
- The function is into (not onto).

Thus, the answer is that \( f \) is neither one-to-one nor onto.

### Final Answer
The function \( f \) is neither one-to-one nor onto.

---